Periodic averaging method for impulsive stochastic dynamical systems driven by fractional Brownian motion under non-Lipschitz condition

Abstract

This paper presents the periodic averaging principle for impulsive stochastic dynamical systems driven by fractional Brownian motion (fBm). Under non-Lipschitz condition, we prove that the solutions to impulsive stochastic differential equations (ISDEs) with fBm can be approximated by the solutions to averaged SDEs without impulses both in the sense of mean square and probability. Finally, an example is provided to illustrate the theoretical results.

Multivalued Impulsive SDEs Driven by G‐Brownian Noise: Periodic Averaging Result

“…Teorem 2 argues that nonautonomous MSDEGs with impulses (18) can strongly be replaced by autonomous MSDEGs without impulses (19). □ Theorem 2.…”

“…□ Theorem 2. Assume Hypotheses 1-4 hold and suppose Z ε and Z ε are solutions for Equations (18) and (19), respectively, then, for any ε 1 ∈ (0, ε 0 ] and 0…”

“…Although more research has been carried out on averaging principles for MSDEs, no controlled studies have been reported for MISDEs. Basically, the idea behind the periodic averaging method for ISDEs is to allow a simplifed autonomous SDE without impulses to replace original nonautonomous ISDEs [17][18][19].…”

Estimating Drift Parameters in a Sub-Fractional Vasicek-Type Process

“…for all t, v ≥ 0. Note that, when H = 1 2 , S H corresponds to the well known Brownian motion B. Sub-fractional Brownian motion has properties that are similar to those of fractional Brownian motion, such as the following: long-range dependence, Self-similarity, Hölder pathes, and it satisfies [17][18][19][20][21][22][23].…”

A Special Study of the Mixed Weighted Fractional Brownian Motion

“…Let ψ ∈ H; then, from [26,27] and following the same technique as used in Section 2 of [25] and Section 2.2 of [28], we obtain…”